## Abstract

Given a field of independent identically distributed (i.i.d.) random variables

*d*-tuples of positive integers and taking values in a separable Banach space

*B*, let

*r*-th maximum of

## Abstract

Given a sequence of identically distributed *ψ*-mixing random variables {*X*
_{n}; *n* ≧ 1} with values in a type 2 Banach space *B*, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.

## Abstract

Let {*X*,*X*_{n}; *n*≧1} be a sequence of *B*-valued i.i.d. random variables. Denote if ∥*X*_{m}∥ is the *r*-th maximum of {∥*X*_{k}∥; *k*≦*n*}, and let be the trimmed sums, where . Given a sequence of positive constants {*h*(*n*), *n*≧1}, which is monotonically approaching infinity and not asymptotically equivalent to loglog*n*, a limit result for is derived.

Considering a simple symmetric random walk in dimension
*d*
≧ 3, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

## Abstract

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.

## Abstract

We study necessary and sufficient conditions for the almost sure convergence of averages of independent random variables with multidimensional indices obtained by certain summability methods.

## Abstract

Let {*X*,*X*_{n}; *n*≧0} be a sequence of identically distributed *ψ*-mixing dependent random variables taking values in a type 2 Banach space *B* with topological dual *B*^{∗}. Considering the geometrically weighted series for 0<*β*<1, a general law of the iterated logarithm for *ξ*(*β*) is obtained without second moment.

## Abstract

Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations
of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any
one of *n* cooperative gamblers who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms
and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic
distributions for the total winnings of the *n* players and from their pooling strategy, where the classes themselves are determined by the two parameters of the game. For
all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that
for a subclass of strategies, not containing the averaging uniform strategy, our merging approximations reduce to asymptotic
expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be
numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations
even for very small *n*.

## Abstract

We investigate the repeated and sequential portfolio St. Petersburg games. For the repeated St. Petersburg game, we show an upper bound on the tail distribution, which implies a strong law for a truncation. Moreover, we consider the problem of limit distribution. For the sequential portfolio St. Petersburg game, we obtain tight asymptotic results for the growth rate of the game.

## Abstract

Let {(*X*
_{nk}, 1≤*k*≤*n*),*n*≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Mricz and
Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means.